However, when studying adiabatic quantum mechanics with families of general Hamiltonians, including non-Hermitian ones, another prominent feature arises. Namely, when the Hamiltonian follows a loop in the space of system parameters, the energies may swap places, which renders the evolution non-cyclic. This effect indicates a non-trivial topology of the energy bands; they need not be separated as in the Hermitian case. The interchange of energies can be related to degeneracies in the space of system parameters, which are known as exceptional points (EPs). These were first mentioned, with slightly different meaning, in [1]}, and have been found in many experimental setups, see the overviews in e.g. [2]} and [3]}. Many of these experiments revolve around parity-time or \(\mathcal {PT}\)  symmetric systems. Such systems are typically described by non-Hermitian Hamiltonians, and the breaking of this symmetry is often associated with an EP. Although \(\mathcal {PT}\) -symmetry is not necessary for EP theory, it does provide experimentally accessible realizations.
